That functionality is enabled by default. It can be disabled by switching off the "Save checklist data to Jira custom fields" toggle in Global Settings page.
When "Save checklist data to Jira custom fields" option is enabled, then checklist changes are recorded in user's Notifications area as presented on the screenshot:
Changes are tracked as the Issue Checklist user. If you would like to see a real user in the issue history, please let us know.